Check the color mode and be sure it’s RGB and not CMYK. The layers need to be at the root level in the file, not within another layer, for AE to read them as separate elements when you import as a comp. When you import it you should have a composition, plus a folder with all of the layers, which you can double-click to open in a footage window to view. The small file size is totally normal since vector files tend to be much smaller than a similarly sized raster file like a psd. HTH